Thanks for your advice! I’ve checked all the steps on the page of your link, except the ”Open a secure web site from your iOS device to see if you can access ports 80 and 443”, which I don't understand and the link doesn't explain. My starting point is that it worked perfectly before I logged out from iCloud and then back in again, all other settings unchanged.
I have also noticed that I had to redo my fingerprints for Touch ID, and having done that some apps don't offer Touch ID anymore but ask me for my password instead. Any connection, do you think?
Sorry about the iOS version, I thought that was automatically inserted in the forum message. I have the latest verson, 14.5.1, and I run Mojave 10.14.6 on my Mac (not having solved 32/64 bit compatability for all my software yet). The iOS was updated only a few days ago, could that instead of the iCloud log-ut have caused the problem?